Residual submodules of multiplication modules

Majid M. Ali

Abstract: Let $R$ be a commutative ring with identity and $M$ an $R$-module. We introduce and give some properties and characterizations of the concepts of $M$-cancellation, $M$-weak cancellation, $M$-meet principal, and $M$-weak meet principal ideals. We prove that a submodule of a faithful multiplication module is faithful (resp. finitely generated, multiplication, flat, projective, pure, prime) if and only if its residual by a finitely generated faithful multiplication ideal is a faithful (resp. finitely generated, multiplication, flat, projective, pure, prime) submodule.